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Exact Boundary Condition for Semi-discretized Schrödinger Equation and Heat Equation in a Rectangular Domain

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Abstract

A convolution type exact/transparent boundary condition is proposed for simulating a semi-discretized linear Schrödinger equation on a rectangular computational domain. We calculate the kernel functions for a single source problem, and subsequently those over the rectangular domain. Approximate kernel functions are pre-computed numerically from discrete convolutionary equations. With a Crank–Nicolson scheme for time integration, the resulting approximate boundary conditions effectively suppress boundary reflections, and resolve the corner effect. The proposed boundary treatment, with a parameter modified, applies readily to a semi-discretized heat equation.

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Acknowledgements

We would like to thank the anonymous referees for stimulating discussions.

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Authors and Affiliations

  1. Institute of Applied Physics and Computational Mathematics, Beijing, 100088, China

    Gang Pang

  2. Department of Statistics, University of Chicago, Chicago, IL, 60637, USA

    Yibo Yang

  3. HEDPS, CAPT and LTCS, College of Engineering, Peking University, Beijing, 100871, China

    Shaoqiang Tang

Authors
  1. Gang Pang
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  2. Yibo Yang
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Corresponding author

Correspondence to Shaoqiang Tang.

Additional information

This research is partially supported by NSFC under Grant Nos. 11272009, 11502208 and 11521202.

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Pang, G., Yang, Y. & Tang, S. Exact Boundary Condition for Semi-discretized Schrödinger Equation and Heat Equation in a Rectangular Domain. J Sci Comput 72, 1–13 (2017). https://doi.org/10.1007/s10915-016-0344-0

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  • DOI: https://doi.org/10.1007/s10915-016-0344-0

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